Numerical stationary states for nonlocal Fokker-Planck equations via fixed points of consistency maps

TL;DR

This paper introduces a fixed-point numerical method for efficiently computing stationary states of nonlocal Fokker-Planck equations, capable of detecting both stable and unstable solutions without time evolution.

Contribution

It develops a novel fixed-point framework reformulating the PDE as a nonlinear map, enabling stable and unstable state detection and improving accuracy through convolution and quadrature treatments.

Findings

Successfully reproduces known bifurcation diagrams

Detects new bifurcation behaviors in model problems

Validates approach with analytical solutions

Abstract

We propose a fixed-point-based numerical framework for computing stationary states of nonlocal Fokker-Planck-type equations. Instead of discretising the differential operators directly, we reformulate the stationary problem as a nonlinear fixed-point map built from the original PDE and its nonlocal interaction terms, and solve the resulting finite-dimensional problem with a matrix-free Newton-Krylov method. We compare implementations using the analytic Frechet derivative of this map with a simple central-difference approximation. Because the method does not rely on time evolution, it is agnostic to dynamical stability and can detect both stable and unstable stationary states. Its accuracy is determined mainly by the numerical treatment of convolutions and quadrature, rather than by differentiation stencils. We apply the approach to three model problems with linear diffusion, use existing analytical results to verify the outputs, and reproduce known bifurcation diagrams, as well as new bifurcation behaviour not previously observed in this kind of problem.